Chemistry Reference
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The result of this section are slight generalizations and extensions of those pre-
sented in Szidarovszky and Zhao (2004). Some simple results are discussed for
multiproduct linear oligopolies in Okuguchi and Szidarovszky (1999).
4.4
Models with Production Adjustment Costs
This section will consider oligopolies in which the firms experience production
adjustment costs. An early contribution along these lines is Howroyd and Rickard
(1981); more recent work on oligopoly models with production adjustment costs
has been carried out for example by Szidarovszky and Yen (1995) and Schoonbeek
(1997). We will consider only discrete time scales as the continuous time case can
be discussed in an analogous fashion. Just as was the case for the models intro-
duced and discussed by Szidarovszky (1999), Chiarella and Szidarovszky (2008a)
and Zhao and Szidarovszky (2008) consider an N-firm oligopoly without external-
ities, with price function f and cost functions C
k
, and assume that any increase or
decrease in the outputs of the firms comes at some cost. Taking this additional cost
component into account, at time period t
C
1 the profit of firm k can be written as
x
k
f.x
k
C
Q
k
.t
C
1//
C
k
.x
k
/
K
k
.x
k
x
k
.t//;
(4.68)
where K
k
is the additional cost component which depends on the amount x
k
x
k
.t/
of output change from the previous time period, and Q
k
.t
C
1/ is the expectation of
the output of the rest of the industry by firm k. In addition to assumptions (A)-(C)
stated at the beginning of Sect. 2.1 for concave oligopolies assume that K
k
is a twice
continuously differentiable convex function.
Under the above conditions, the expression (4.68) is strictly concave in x
k
and if
each firm has a finite capacity limit, then there is always a unique best response R
k
,
which depends on both Q
k
.t
C
1/ and x
k
.t/ and can be 0;L
k
; or an interior value.
Consider the case of an interior equilibrium. In a small neighborhood of it the
best responses are also interior, and the first order condition implies that
f.x
k
C
Q
k
/
C
x
k
f
0
.x
k
C
Q
k
/
C
k
.x
k
/
K
0
k
.x
k
x
k
.t//
D
0;
where we use the simplifying notation Q
k
for Q
k
.t
C
1/:
The left hand side of the last equation is strictly decreasing in x
k
, so this equation
has a unique solution, x
k
D
R
k
.Q
k
;x
k
.t//, which depends on both Q
k
and x
k
.t/.
By implicit differentiation with respect to Q
k
one has
f
0
.R
0
kQ
C
1/
C
R
0
kQ
f
0
C
x
k
f
00
.R
0
kQ
C
1/
C
0
k
R
0
kQ
K
0
k
R
0
kQ
D
0;
and with respect to x
k
.t/,
f
0
R
0
kx
C
R
0
kx
f
0
C
x
k
f
00
R
0
kx
C
0
k
R
0
kx
K
0
k
.R
0
kx
1/
D
0;
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